Maths Essentials, Unit 18, Lesson 3 of 5

Rates of change from graphs

GCSE Maths, Rates, compound measures and growth, 25 minutes

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The gradient of a graph is a rate; the area under a speed-time graph is a distance; tangents give rates at an instant.

What you will learn

  • Find a speed or acceleration from a gradient
  • Find a distance from the area under a speed-time graph
  • Estimate a rate at an instant from a tangent (Higher)

Gradient is a rate

The gradient of a line is how steep it is: the change up ÷ the change across.

Its meaning comes from the two axes, so it is always "up-axis unit per across-axis unit".

On a distance-time graph the gradient is the speed; on a speed-time graph it is the acceleration; on a pay-hours graph it is the pay per hour. A flat section means nothing is changing: on a distance-time graph that means stopped, and on a speed-time graph it means a steady speed.

  1. a walker's line rises from 0 km at 0 minutes to 3 km at 45 minutes
  2. gradient = 3 ÷ 45 = 1/15 km per minute
  3. × 60 for km/h: 1/15 × 60 = 4 km/h

When the time axis is in minutes, the gradient is per minute: multiply by 60 for per hour. For the journey home, use only the time of that section (end time − start time). When two lines are on the same graph, the steeper line has the greater rate; subtract the two gradients to say how much greater.

Acceleration is the change in speed ÷ the time taken, in m/s². A car going from 4 m/s to 10 m/s in 3 seconds accelerates at (10 − 4) ÷ 3 = 2 m/s². Use the change in speed, not the final speed.

The rest of this lesson and its 7 practice questions are locked.

In this lesson

  1. Gradient is a rate
  2. Filling containers
  3. Area is distance
  4. Curved graphs (Higher)
  5. Try it: Gradients on the graphing calculator

7 practice questions, marked as you go; about 25 minutes in all.

Key terms

Gradient, Steeper, Acceleration, Narrower, Wider, Tangent, Average rate

In this unit: Rates, compound measures and growth

Speed, density and pressure, rates of change read from graphs, compound interest and depreciation, and exponential growth and decay.

All of Rates, compound measures and growth

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